Group #28377986 — subgroups

Subgroups — #28377986

Order 13952.

110 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295912	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295889	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182032	—	2	1	no	—
5	C₄⋊#11724786	—	2	1	no	—
6	D₄×Dic₁₀₉	—	2	1	no	—
7	#16812633	C₄	1	1	yes	characteristic
8	#16812620	—	2	1	no	—
9	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#11724786	—	4	1	no	—
12	#13182032	—	2	2	no	—
13	C₂²×Dic₁₀₉	—	4	1	no	—
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	—	2	1	no	—
16	C₂²⋊Dic₁₀₉	—	4	1	no	—
17	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₀₉	—	2	1	no	—
19	D₈×C₁₀₉	—	2	1	no	—
20	D₈×C₁₀₉	C₈	1	2	yes	complemented
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₁₀₉	—	4	1	no	—
23	D₄×C₁₀₉	—	2	2	no	—
24	D₄×C₁₀₉	—	4	1	no	—
25	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₀₉	—	4	1	no	—
27	C₂×Dic₁₀₉	—	8	1	no	—
28	C₂×Dic₁₀₉	—	8	1	no	—
29	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
30	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
31	#11724786	—	8	1	no	—
32	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
34	#11724784	—	2	1	no	—
35	C₂²×C₁₀₉	—	4	2	no	—
36	C₂²×C₁₀₉	—	8	1	no	—
37	C₂²×C₁₀₉	—	8	1	no	—
38	C₂²×C₁₀₉	#102	1	1	yes	characteristic
39	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₁₀₉	#104	1	1	yes	characteristic
41	Dic₁₀₉	—	2	1	no	—
42	Dic₁₀₉	—	8	1	no	—
43	Dic₁₀₉	D₁₆	1	1	yes	characteristic
44	Dic₁₀₉	SD₃₂	1	1	yes	characteristic
45	C₂×C₁₀₉	—	8	2	no	—
46	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
47	C₂×C₁₀₉	#357	1	1	yes	characteristic
48	C₂×C₁₀₉	#359	1	1	yes	characteristic
49	#1331	—	109	1	no	maximal, Sylow
50	C₁₀₉	#1331	1	1	yes	characteristic, complemented, Sylow
51	C₄×D₈	—	109	1	no	—
52	C₄×C₁₆	—	109	1	no	—
53	#335	—	109	1	no	—
54	C₂×D₈	#89137	1	1	yes	characteristic
55	C₄×D₄	—	218	1	no	—
56	C₂×C₁₆	—	218	1	no	—
57	#107	—	109	1	no	—
58	C₄⋊C₈	—	218	1	no	—
59	#102	—	218	1	no	—
60	C₄×C₈	—	109	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂²×C₄	—	436	1	no	—
63	D₈	—	2	1	no	—
64	D₈	#11724786	1	2	yes	complemented
65	C₂×C₈	—	109	1	no	—
66	C₂×C₈	—	436	1	no	—
67	C₂×C₈	C₂×#89137	1	1	yes	characteristic
68	C₄⋊C₄	—	218	1	no	—
69	C₂²⋊C₄	—	436	1	no	—
70	C₄²	—	109	1	no	—
71	C₁₆	—	218	2	no	—
72	C₂³	—	4	1	no	—
73	D₄	—	2	2	no	—
74	D₄	—	4	1	no	—
75	C₂×C₄	—	872	1	no	—
76	C₂×C₄	—	872	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	1	no	—
79	C₂×C₄	—	436	1	no	—
80	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
81	C₈	—	872	1	no	—
82	C₈	—	218	1	no	—
83	C₈	C₂×#11724786	1	1	yes	characteristic
84	C₈	#13182038	1	1	yes	characteristic
85	C₂²	—	4	2	no	—
86	C₂²	—	8	1	no	—
87	C₂²	—	8	1	no	—
88	C₂²	#16812761	1	1	yes	characteristic
89	C₄	—	872	1	no	—
90	C₄	—	109	1	no	—
91	C₄	—	109	1	no	—
92	C₄	—	218	1	no	—
93	C₄	C₂²⋊#11724786	1	1	yes	characteristic
94	C₄	#16812763	1	1	yes	characteristic
95	C₂	—	8	2	no	—
96	C₂	D₄⋊#11724786	1	1	yes	characteristic
97	C₂	#20295938	1	1	yes	characteristic
98	C₂	#20295940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295912

C₂

yes

maximal, characteristic, complemented

#20295889

C₂

yes

maximal, characteristic, complemented

D₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182032

—

C₄⋊#11724786

—

D₄×Dic₁₀₉

—

#16812633

C₄

yes

characteristic

#16812620

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

#13182032

—

C₂²×Dic₁₀₉

—

C₂×#11724784

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

C₈

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₈×C₁₀₉

M₁₆

yes

characteristic

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#102

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₄×C₁₀₉

#104

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

D₁₆

yes

characteristic

Dic₁₀₉

SD₃₂

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

D₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1331

yes

characteristic, complemented, Sylow

C₄×D₈

—

109

—

C₄×C₁₆

—

109

—

#335

—

109

—

C₂×D₈

#89137

yes

characteristic

C₄×D₄

—

218

—

C₂×C₁₆

—

218

—

#107

—

109

—

C₄⋊C₈

—

218

—

#102

—

218

—

C₄×C₈

—

109

—

C₂×D₄

—

C₂²×C₄

—

436

—

D₈

—

D₈

#11724786

yes

complemented

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

436

—

C₄²

—

109

—

C₁₆

—

218

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

872

—

C₂×C₄

—

872

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₂²⋊#89137

yes

characteristic

C₈

—

872

—

C₈

—

218

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#16812761

yes

characteristic

C₄

—

872

—

C₄

—

109

—

C₄

—

109

—

C₄

—

218

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#16812761
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting, prime radical	C₂×D₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#102
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	C₂×D₈	#89137
hypocenter	C₁₀₉	#1331
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂²

#16812761

commutator

C₈×C₁₀₉

C₂×C₈

Frattini

C₂×C₈

C₂×#89137

Fitting, prime radical

C₂×D₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#102

cosocle

C₈×Dic₁₀₉

C₂²

C₂×D₈

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #20295912	C₂	semidirect
2. #20295889	C₂	semidirect
20. D₈×C₁₀₉	C₈	semidirect
50. C₁₀₉	#1331	semidirect
64. D₈	#11724786	semidirect

H ≅ G/N

type

1. #20295912

C₂

semidirect

2. #20295889

C₂

semidirect

20. D₈×C₁₀₉

C₈

semidirect

50. C₁₀₉

#1331

semidirect

64. D₈

#11724786

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295912	2
#20295889	2
D₈×Dic₁₀₉	2
#1331	109

subgroup

index

#20295912

#20295889

D₈×Dic₁₀₉

#1331

109

Subgroups — #28377986

Conjugacy classes of subgroups

Complements

Maximal subgroups